Approximate Polynomial Satisfiability is in the Counting Hierarchy

2026-09-30

AI Summary

This study addresses the problem of excessively high upper bounds on the computational complexity of Approximate Polynomial Satisfiability (APS). Methodologically, it integrates counting hierarchy theory with algebraic geometry tools. Building upon recent results from FOCS 2026, this work establishes a reduction between APS and Hilbert’s Nullstellensatz, proving for the first time that APS belongs to the Counting Hierarchy (CH). The core contributions are threefold: significantly reducing the complexity of APS from PSPACE to CH, formally establishing its membership in CH, and optimizing the hitting set certification complexity for border classes to the CH level. Consequently, this research systematically improves the theoretical bounds for multiple related problems within algebraic complexity theory.

Abstract

The Approximate polynomial satisfiability problem (APS), introduced by Guo, Saxena, and Sinhababu (CCC 2018), asks whether the zero vector lies in the Zariski closure of the image of a given polynomial map. Specifically, for a field $k$ with algebraic closure~$K$, the problem asks whether $\boldsymbol 0 \in\overline{\boldsymbol f(K^n)}$ for a polynomial map $\boldsymbol f=(f_1,\ldots,f_m)$ with $f_i\in k[X_1,\ldots,X_n]$. APS is a natural topological analogue of Hilbert's Nullstellensatz, namely the question of whether a given system of polynomial equations has a common zero. APS captures several problems in algebraic complexity, including border rank, hitting sets for border classes, and null-cone membership; it is known to be NP-hard and in PSPACE. We show that APS lies in the Counting Hierarchy (CH) over both the rationals and finite fields, substantially improving the known PSPACE upper bound. Our proof builds on a recent breakthrough due to Andrews, Garg, and Schost (FOCS 2026) on deciding Hilbert's Nullstellensatz in CH. As a corollary, our result improves the complexity of certifying hitting sets for border classes from PSPACE to CH. We also give a polynomial-time reduction of Hilbert's Nullstellensatz to APS, valid in any characteristic. In characteristic zero, we give a reduction of APS to the decision problem for the existential theory of real closed fields. Overall, our results place approximate polynomial satisfiability closer in complexity to exact polynomial feasibility and as a byproduct give improved complexity bounds for several problems arising in approximative complexity.

Problem

• Approximate Polynomial Satisfiability
• Computational Complexity
• Counting Hierarchy
• Algebraic Complexity
• Hilbert's Nullstellensatz

Innovation

• Approximate Polynomial Satisfiability
• Counting Hierarchy
• Hilbert's Nullstellensatz
• Algebraic Complexity
• Polynomial-time Reduction